Free modal algebras revisited
... methodology used for introducing a logical calculus. With a clear combinatorial and conceptual description of free algebras in mind, one can better investigate metatheoretical properties like admissibility of inference rules, solvability of equations, definability and interpretability matters, etc. ...
... methodology used for introducing a logical calculus. With a clear combinatorial and conceptual description of free algebras in mind, one can better investigate metatheoretical properties like admissibility of inference rules, solvability of equations, definability and interpretability matters, etc. ...
Lie groups, 2013
... See Helgason’s book, pp 339-347 (first edition) for a more complete list and descriptions. The fact that all the above are actually submanifolds can be shown directly, but it is interesting that this is also a consequence of a general theorem about algebraic groups, stated now as our first problem. ...
... See Helgason’s book, pp 339-347 (first edition) for a more complete list and descriptions. The fact that all the above are actually submanifolds can be shown directly, but it is interesting that this is also a consequence of a general theorem about algebraic groups, stated now as our first problem. ...
Vectors-with
... We can use either one to solve for θ. We will get the same answer. The cosine equation turns out to be the easier equation to work with when it comes to finding the direction of a vector. If you want to see how to find the direction by solving the sine equation, go to slide 30 of this slide show. ...
... We can use either one to solve for θ. We will get the same answer. The cosine equation turns out to be the easier equation to work with when it comes to finding the direction of a vector. If you want to see how to find the direction by solving the sine equation, go to slide 30 of this slide show. ...
slides
... • The result of the stretching is exactly the same as transformation by a matrix • The axes of stretching/shrinking are the eigenvectors – The degree of stretching/shrinking are the corresponding eigenvalues • The EigenVectors and EigenValues convey all the information about the matrix ...
... • The result of the stretching is exactly the same as transformation by a matrix • The axes of stretching/shrinking are the eigenvectors – The degree of stretching/shrinking are the corresponding eigenvalues • The EigenVectors and EigenValues convey all the information about the matrix ...
Vector Spaces
... force, called the resultant or sum, acted on the object, we define the sum of two vectors in order to model how forces combine. Thus, if we want the vector A + B , we draw a dashed line segment starting at the tip of A , parallel to B , with the same length as B ; cf. Figure 2.2a and b. Label the en ...
... force, called the resultant or sum, acted on the object, we define the sum of two vectors in order to model how forces combine. Thus, if we want the vector A + B , we draw a dashed line segment starting at the tip of A , parallel to B , with the same length as B ; cf. Figure 2.2a and b. Label the en ...
DUAL MODULES 1. Introduction
... ci f (ei ) = (c1 , . . . , cn ) · (f (e1 ), . . . , f (en )) = ϕv (w) where v = (f (e1 ), . . . , f (en )). So f = ϕv for this choice of v. The fact that Rn can be identified with (Rn )∨ using the dot product may have delayed somewhat the development of abstract linear algebra, since it takes a cert ...
... ci f (ei ) = (c1 , . . . , cn ) · (f (e1 ), . . . , f (en )) = ϕv (w) where v = (f (e1 ), . . . , f (en )). So f = ϕv for this choice of v. The fact that Rn can be identified with (Rn )∨ using the dot product may have delayed somewhat the development of abstract linear algebra, since it takes a cert ...
Euclidean Spaces
... that is, if these vectors are mutually orthogonal and the length of each of them is equal to 1. If m = n and the vectors e1 , . . . , en form a basis of the space, then such a basis is called an orthonormal basis. It is obvious that the Gram determinant of an orthonormal basis is equal to 1. We shal ...
... that is, if these vectors are mutually orthogonal and the length of each of them is equal to 1. If m = n and the vectors e1 , . . . , en form a basis of the space, then such a basis is called an orthonormal basis. It is obvious that the Gram determinant of an orthonormal basis is equal to 1. We shal ...
Lecture notes Math 4377/6308 – Advanced Linear Algebra I
... 4. additive inverses: for every x ∈ X there exists (−x) ∈ X such that x + (−x) = 0; 5. associativity of multiplication: a(bx) = (ab)x for all a, b ∈ R and x ∈ X; 6. distributivity: a(x+y) = ax+ay and (a+b)x = ax+bx for all a, b ∈ R and x, y ∈ X; 7. multiplication by the unit: 1x = x for all x ∈ X. T ...
... 4. additive inverses: for every x ∈ X there exists (−x) ∈ X such that x + (−x) = 0; 5. associativity of multiplication: a(bx) = (ab)x for all a, b ∈ R and x ∈ X; 6. distributivity: a(x+y) = ax+ay and (a+b)x = ax+bx for all a, b ∈ R and x, y ∈ X; 7. multiplication by the unit: 1x = x for all x ∈ X. T ...
... covariant and contravariant components of vectors, and spherical harmonics. After laying the necessary linear algebraic foundations, we give in Chap. 3 the modern (component-free) definition of tensors, all the while keeping contact with the coordinate and matrix representations of tensors and their ...
NOTES ON LINEAR ALGEBRA
... should have the same number of rows and columns. They should have the same number of columns because they both act on the same vector. They should have the same number of rows because they should each take that vector to the same space. For example, here’s an example of what can go wrong when we try ...
... should have the same number of rows and columns. They should have the same number of columns because they both act on the same vector. They should have the same number of rows because they should each take that vector to the same space. For example, here’s an example of what can go wrong when we try ...
Frobenius monads and pseudomonoids
... recall the basic concept. The connection between TQFT and Frobenius algebras is pointed out in [Ko] and we proceed to outline how that works. Some connection between quantum groups and Frobenius algebras is already apparent from the fact that quantum groups are Hopf algebras and finite-dimensional H ...
... recall the basic concept. The connection between TQFT and Frobenius algebras is pointed out in [Ko] and we proceed to outline how that works. Some connection between quantum groups and Frobenius algebras is already apparent from the fact that quantum groups are Hopf algebras and finite-dimensional H ...
Math 217: Multilinearity of Determinants Professor Karen Smith A
... Solution note: The rank is 3, which is less than 6. So the determinant is 0. (3) A basis for the image is represented by a maximal set of linearly independent columns. These can be taken to be the first, second and third columns. So a basis for the image is the set (A3 , A1 − A2 − A4 − A5 − A6 , A1 ...
... Solution note: The rank is 3, which is less than 6. So the determinant is 0. (3) A basis for the image is represented by a maximal set of linearly independent columns. These can be taken to be the first, second and third columns. So a basis for the image is the set (A3 , A1 − A2 − A4 − A5 − A6 , A1 ...
Linear Algebra Abridged - Linear Algebra Done Right
... x C . x/ D 0: In other words, if x D .x1 ; : : : ; xn /, then x D . x1 ; : : : ; xn /. For a vector x 2 R2 , the additive inverse x is the vector parallel to x and with the same length as x but pointing in the opposite direction. The figure here illustrates this way of thinking about the additive in ...
... x C . x/ D 0: In other words, if x D .x1 ; : : : ; xn /, then x D . x1 ; : : : ; xn /. For a vector x 2 R2 , the additive inverse x is the vector parallel to x and with the same length as x but pointing in the opposite direction. The figure here illustrates this way of thinking about the additive in ...
Lie algebras of locally compact groups
... and 3 we prove the existence and uniqueness of the Lie algebra of an LP-group and show the connection of the Lie algebra with the group by means of the exponential mapping. In § 4, we extend the notion of a universal covering group for connected groups with the same Lie algebra. A covering group of ...
... and 3 we prove the existence and uniqueness of the Lie algebra of an LP-group and show the connection of the Lie algebra with the group by means of the exponential mapping. In § 4, we extend the notion of a universal covering group for connected groups with the same Lie algebra. A covering group of ...
(Non-)Commutative Topology
... discrete topology nor with the counting measure. Instead, non-trivial topology and measure theory will be necessary. The framework is the theory of commutative C∗ -algebras (“C-star-algebra”), an extremely beautiful branch of functional analysis. In essence, this theory boils down to the following: ...
... discrete topology nor with the counting measure. Instead, non-trivial topology and measure theory will be necessary. The framework is the theory of commutative C∗ -algebras (“C-star-algebra”), an extremely beautiful branch of functional analysis. In essence, this theory boils down to the following: ...
MATH 110 Midterm Review Sheet Alison Kim CH 1
... then no linear map from V to W is surj calculating a matrix: let T ∈ L(V,W). suppose (v1,…,vn) is a basis of V and (w1,…,wm) is a basis of W. for each k=1,…,n, we can write Tvk uniquely as a linear combination of w’s: Tvk=a1,kw1+…+am,kwm | aj,k ∈ F for j=1,…,m. then matrix is given by M(T,(v1,…,vn), ...
... then no linear map from V to W is surj calculating a matrix: let T ∈ L(V,W). suppose (v1,…,vn) is a basis of V and (w1,…,wm) is a basis of W. for each k=1,…,n, we can write Tvk uniquely as a linear combination of w’s: Tvk=a1,kw1+…+am,kwm | aj,k ∈ F for j=1,…,m. then matrix is given by M(T,(v1,…,vn), ...
linear algebra - Universitatea "Politehnica"
... (i) K as a vector space over itself, with addition and multiplication in K. (ii) Kn is a K–vector space. (iii) Mm,n (K) with usual matrix addition and multiplication by scalars is a K–vector space. (iv) The set of space vectors V3 is a real vector space with addition given by the parallelogram law a ...
... (i) K as a vector space over itself, with addition and multiplication in K. (ii) Kn is a K–vector space. (iii) Mm,n (K) with usual matrix addition and multiplication by scalars is a K–vector space. (iv) The set of space vectors V3 is a real vector space with addition given by the parallelogram law a ...
linear algebra - Math Berkeley - University of California, Berkeley
... are characterized by their magnitude and direction. Yet, the popular slogan “Vectors are magnitude and direction” does not qualify for a mathematical definition of vectors, e.g. because it does not tell us how to operate with them. The computer science definition of vectors as arrays of numbers, to ...
... are characterized by their magnitude and direction. Yet, the popular slogan “Vectors are magnitude and direction” does not qualify for a mathematical definition of vectors, e.g. because it does not tell us how to operate with them. The computer science definition of vectors as arrays of numbers, to ...
FUZZY SEMI-INNER-PRODUCT SPACE Eui-Whan Cho, Young
... kxk = |bx · xc| 2 with values in R∗ (I). We write (X, ·, k k) to show that the norm k k is the function thus derived from the fuzzy semi-inner-product. Let us new define the real quadratic form <, >α : X × X → R for every x, y ∈ X,α ∈ I1 fixed, by < x, y >α = (x · y)(α) . Lemma 2.5. If for α ∈ I1 an ...
... kxk = |bx · xc| 2 with values in R∗ (I). We write (X, ·, k k) to show that the norm k k is the function thus derived from the fuzzy semi-inner-product. Let us new define the real quadratic form <, >α : X × X → R for every x, y ∈ X,α ∈ I1 fixed, by < x, y >α = (x · y)(α) . Lemma 2.5. If for α ∈ I1 an ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.